A couple of years ago, Maryla Smolicz's Polish deli in Swindon was besieged with new arrivals seeking the comforts of imported kielbasa and advice on setting up life in the southern English town. They came to Britain because their earnings would far exceed what they made in Poland. "Now, they're all going home," says Smolicz. "This week alone, two couples have stopped by to thank me and say goodbye."

Since the European Union's enlargement in 2004, when Britain opened its job market to Europe's new member states, Poles have provided the British economy with a...